Related papers: Projective Wonderful Models for Toric Arrangements
We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We…
By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $\Omega^U_*\simeq \mathbb Z[a_1,a_2,\dots]$, ${\rm deg}(a_i)=2i$. In this paper we solve a…
We provide an algorithmic method for constructing projective resolutions of modules over quotients of path algebras. This algorithm is modified to construct minimal projective resolutions of linear modules over Koszul algebras.
The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give…
We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in…
Inspired by a paper of Salberger we give a new proof of Manin's conjecture for toric varieties over imaginary quadratic number fields by means of universal torsor parameterizations and elementary lattice point counting.
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
We completely describe the Fano scheme of lines for a projective toric surface in terms of the geometry of the corresponding lattice polygon.
We describe the intersection complex of any perversity of a toric variety completely in terms of the associated fan. It is described by a finite complex of finite dimensional graded vector spaces which we call graded exterior modules. The…
We study the enumerative geometry of orbits of multidimensional toric action on projective algebraic varieties and develop a new cyclic differential-graded operad, conjecturally governing the real version of the enumerative geometry of…
In this paper, we obtain a complete classification of smooth toric Fano varieties equipped with extremal contractions which contract divisors to curves for any dimension. As an application, we obtain a complete classification of smooth…
The derived category of an algebraic variety might be a source of a myriad of new (categorical) symmetries. Some are predicted by homological mirror symmetry, to be obtained from the fundamental group of the space of complex structures of…
We construct a full strong exceptional collection consisting of line bundles on any two-dimensional smooth toric weak Fano stack. The total endomorphism algebra of the resulting collection is isomorphic to the path algebra of a quiver with…
We generalize classical results about the topology of toric varieties to the case of projective Q-factorial T-varieties of complexity one using the language of divisorial fans. We describe the Hodge-Deligne polynomial in the smooth case,…
By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising…
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; using Mirror Symmetry these…
Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi-Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated…
This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular,…
Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an…
Exceptional sequences of line bundles on a smooth projective toric surface are automatically full when they can be constructed via augmentation. By using spherical twists, we give examples that there are also exceptional sequences which can…